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Simulation and Analysis of Ultrasonic Wave

Propagation in Pre-stressed Screws

Erik Andrén

Engineering Physics and Electrical Engineering, master's level 2019

Luleå University of Technology

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A

BSTRACT

The use of ultrasound to measure preload in screws and bolts has been studied quite frequently the last decades. The technique is based on establishing a relationship between preload and change in time of flight (TOF) for an ultrasonic pulse propagating back and forth through a screw. This technique has huge advantages compared to other methods such as torque and angle tightening, mainly because of its independence of friction. This is of great interest for Atlas Copco since it increases the accuracy and precision of their assembly tools.

The purpose of this thesis was to investigate ultrasonic wave propagation in pre-stressed screws using a simulation software, ANSYS, and to analyse the results using signal pro-cessing. The simulations were conducted in order to get an understanding about the wavefront distortion effects that arise. Further, an impulse response of the system was estimated with the purpose of dividing the multiple echoes that occur from secondary propagation paths from one other.

The results strengthen the hypothesis that the received echoes are superpositions of reflections taking different propagation paths through the screw. An analytical estimation of the wavefront curvature also shows that the wavefront distortion due to a higher stress near the screw boundaries can be neglected. Additionally, a compressed sensing technique has been used to estimate the impulse response of the screw. The estimated impulse response models the echoes as superpositions of secondary echoes, with significant taps corresponding to the TOF of the shortest path and a mode-converted echo. The method is also shown to be stable in noisy environments.

The simulation model gives rise to a slower speed of sound than expected, which most likely is due to the fact that finite element analysis in general overestimates the stiffness of the model.

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P

REFACE

This thesis has been written with the purpose of completing my Master’s degree in En-gineering Physics and Electrical EnEn-gineering at Lule˚a University of Technology (LTU). I am very thankful to have been given this opportunity working at Atlas Copco, a company with fantastic co-workers and a great working environment.

My supervisors at Atlas Copco have been invaluable during the work; Erik Persson for his wide expertise in all fields, and Johan N˚asell for his talent in explaining and guiding me through many obstacles on the way.

Equally, I would like to thank my supervisor at LTU, Professor Johan Carlson, for sharing his research and ideas in this field, always challenging me in a way where the learning objective and my personal development are in focus.

A special thanks goes to Andreas Rydin at ANSYS Sweden AB for his professional support and feedback, which in the end made the simulation results possible at all.

Finally, thanks again to Johan Carlson for letting me use his stylish LaTeX template during the writing of this thesis.

Enjoy reading!

Erik Andr´en 16th June 2019

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C

ONTENTS

Nomenclature ix Chapter 1 – Introduction 1 1.1 Background . . . 1 1.2 Screw Joints . . . 2 1.3 Tightening . . . 3 1.3.1 Torque Tightening . . . 3 1.3.2 Ultrasonic Measurements . . . 3 1.4 Objective . . . 5 1.5 Limitations . . . 5 1.6 Outline . . . 5 Chapter 2 – Theory 7 2.1 Solid Mechanics . . . 7 2.1.1 Hooke’s Law . . . 7 2.1.2 Elastic Deformations . . . 8

2.2 Finite Element Method (FEM) . . . 10

2.2.1 Mode Superposition (MSUP) Method . . . 10

2.2.2 Full Method . . . 11

2.3 Ultrasonic Wave Propagation . . . 11

2.3.1 Attenuation . . . 11

2.3.2 Wave Modes . . . 12

2.3.3 Reflection and Mode Conversion . . . 12

2.3.4 Piezoelectric Transducers . . . 14

2.3.5 Wavefront Distortion . . . 14

2.4 Acoustoelastic Effect . . . 17

2.4.1 Small-on-large Theory . . . 17

2.4.2 The Plane Wave . . . 18

2.5 Ultrasonic Preload Measurements . . . 20

2.6 Compressed Sensing . . . 21

Chapter 3 – Method 25 3.1 Finite Element Analysis . . . 25

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3.1.2 Transient Analysis . . . 26

3.1.3 Pre-stressed Transient Analysis . . . 28

3.2 Post-processing . . . 29 3.2.1 Wavefront Curvature . . . 29 3.2.2 Compressed Sensing . . . 29 Chapter 4 – Results 31 4.1 Wave Propagation . . . 31 4.2 Secondary Echoes . . . 31

4.3 Pre-stressed Transient Analysis . . . 34

4.4 Wavefront Curvature . . . 36

4.5 Compressed Sensing . . . 36

Chapter 5 – Discussion & Conclusion 41 5.1 Interpretation of Results . . . 41

5.2 Simulation Limitations . . . 42

5.3 Conclusion . . . 44

5.4 Future Work . . . 44

Appendix A – ISO Screw Dimensions 45

Appendix B – Mode-converted Echo 47

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Nomenclature

Acronyms

∆TOF Difference in time of flight AWGN Additive white Gaussian noise CAD Computer-aided design

CS Compressed sensing DOF Degrees of freedom FEA Finite element analysis FEM Finite element method

ISO International Organization for Stan-dardization

LASSO Least absolute shrinkage and selec-tion operator

LS Least-squares

MSUP Mode superposition NDT Non-destructive testing

ODE Ordinary differential equation RR Ridge regression

SNR Signal-to-noise ratio TOF Time of flight US Ultrasonic Symbols αq Amplitude χ Mapping function ∆x Element size δ Elongation `1, `2 Norm functions κ Bulk modulus λ, µ Lam´e constants Λ Acoustic tensor C, ξi Damping coefficients

F Deformation gradient tensor F Nodal acting forces

I Identity matrix

K Global stiffness matrix

M Mass

m Polarization direction n Propagation direction S Nominal stress tensor U Convolution matrix u Input signal

u Nodal displacements ix

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x, X Position vectors y Recorded displacement µR, γ Regularization parameters ν Poission’s ratio ω Frequency φi Mode shapes ρ Density σ Tensile load τ Time delay ε Cauchy strain A cross sectional area

A, B, C Third order elastic constants Aii(z) Acoustoelastic constant

aij Acoustoelastic coefficients cl, cii(z) Longitudinal wave speed cp Phase velocity

cs, cij(z) Transversal wave speed D Nominal diameter

E Young’s modulus e(t) Random noise

f Twice differentiable function Fpre Preload H(ω), h(t), h Transfer function J Jacobian j Imaginary unit, √−1 K Screw stiffness k Nut factor

k(ω) Complex wave number L Length of screw

l, m, n Murnaghan constants P Acoustic pressure P, p Material points

Q Number of propagation paths q Propagation path q R Reflection coefficient T Torque t Time W Strain energy yi Modal amplitudes Z Acoustic impedance x

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C

HAPTER

1

Introduction

This chapter will give some background and purpose to ultrasonic measurements for determining the preload in screw joints. Thereafter, the objective, limitations and an outline of the thesis will be given.

1.1

Background

The industrial company Atlas Copco is a world leading provider of sustainable productivity-enhancing solutions. Their business area in industrial technique is mainly providing power tools and assembly systems [1, 2]. The by far most common method when se-curing parts and components to one other is to use a screw and a nut (or a threaded hole) to clamp the joint members together. The reason for this is its simplicity and cost efficiency. When tightening a screw joint, the force with which it is tightened is known as the preload. The accuracy of the preload is essential for the screw joint to work properly. Some of the factors that affect the accuracy of the preload are: tools, operator, control method, short and long-term relaxation, external loads, and the quality of parts. This thesis will look at improvements of the control method, i.e. measuring the preload. Most tightening techniques are very sensitive to variation in friction in the contact areas of the screw joint, resulting in a poor accuracy. Atlas Copco is therefore investigating the possibility of measuring the preload with ultrasonic pulses, a technique independent of friction. The time of flight (TOF) for an ultrasonic pulse is defined as the time it takes for a sound wave to propagate back and forth through the screw. As the screw is tight-ened the TOF will increase; partially from the fact that it gets longer, but also due to a decrease of sound speed (a phenomena known as the acoustoelastic effect). Thus the preload can be estimated very accurately by measuring the difference in TOF (∆TOF) between an untightened and tightened screw. Previous studies have proven this by fitting a mathematical model to measured data. However, when the clamp length could not be assumed to be much longer than the diameter of the screw the one-dimensional analysis

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2 Introduction

failed [3]. Further, the ultrasonic pulse suffers from distortion, resulting in difficulties in detecting the received pulse and its TOF. Quantization errors will also arise due to a limited sampling rate.

Martinsson et al have presented a parametric method for determining the frequency dependent attenuation and phase velocity in dispersive media using ultrasonic pulse-echo measurements [4, 5]. This method is based on finding a parametric model for the fre-quency dependent transfer function of the system. Unlike cross-correlation methods (like TOF measurements) the method is unbiased, and the noise is significantly suppressed compared to non-parametric methods such as Fourier analysis. Continuing with a similar approach, Carlson et al have used a compressed sensing technique to find the impulse response from overlapping ultrasonic echoes when investigating thin materials [6]. This can be compared to the overlapping of echoes that results from sidewall reflections and secondary propagation paths in a screw.

1.2

Screw Joints

The screw joint to be considered in this work is depicted in Fig. 1.1, which consists of a screw and a nut, holding two different parts together. In this instance two metal plates. The screws that will be used are the ISO-standard M8 and M10, with diameters of 8 and 10 mm [7]. The length of the screws will be in the region of 70-100 mm. The relation between thread height and pitch can be found in Appendix A.

Tensile load Clamping force

Tensile load

Shear load Shear load

Figure 1.1: A screw joint consisting of a partly threaded screw and a nut holding two different parts together. The clamping force, shear load, and tensile load are shown in the figure [8].

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1.3. Tightening 3

1.3

Tightening

The purpose of tightening is to achieve the right clamp force in a screw joint for it to work properly. The clamping force is the force holding the joint members together. Further, the preload is defined as the clamping force right after the screw joint is tightened. To make sure that the clamped parts do not separate from one other, the clamping force should be greater than the maximum load that the joint will be exposed to. When a load is applied, the clamping force will no longer remain equal to the preload. Hence, we always want the maximum possible preload, where as a rule of thumb the minimum allowed preload is to be greater than twice the maximum load that the joint will carry [9]. A screw joint is tightened by applying a torque to either the head or the nut. About 90% of the torsional work will be consumed by friction in the head, nut and threads. Thus only 10% of the torque will be converted into axial force - tension of the screw [10]. Additionally, the friction varies a lot between different screw joints.

The preload during tightening can be calculated using one of numerous different tech-niques. The method to be chosen depends on the purpose of the screw joint. Typically, the more expensive methods give a higher accuracy of preload. Afterwards, the preload can be verified using a method of more accurate measurement, for an instance, using ultrasound. Below, the torque tightening technique followed by ultrasonic measurements to determine preload will be described. For additional tightening techniques and methods for measuring preload the reader is referred to the thesis written by P¨arlstrand [3].

1.3.1

Torque Tightening

Torque tightening is performed by applying a specific torque to a screw joint, e.g. by using a torque wrench. For an ISO standard screw, the applied torque, T , is chosen from a desired preload, Fpre, such that [11]

T = kFpreD, (1.1)

where D is the nominal screw diameter, and k is the nut factor given from the properties of the screw; thread pitch, thread angle, and friction coefficients in the threads, head and nut. As already mentioned, this method has a huge drawback. Since it is very friction dependent, the 3σ level could reach 40-60% of the preload [12].

1.3.2

Ultrasonic Measurements

Ultrasonic measurements exploit the fact that the screw gets elongated due to an applied preload. An ultrasonic transducer is used to perform a pulse-echo measurement, i.e. transmit an ultrasonic pulse and record its echoes. The procedure of an ultrasonic pulse-echo measurement is illustrated in Fig. 1.2. Thus, if one could determine the elongation, δ, the preload (considering only one degree of freedom) would simply be obtained as [13]

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4 Introduction

US Transducer

𝛿 Transmitted pulse

Received echo

Figure 1.2: Ultrasonic (US) pulse-echo measurement to determine the elongation, δ, of a screw subject to an applied preload, Fpre.

where K is the total axial screw stiffness. Stiffness is a property of the screw, and can be approximated as the sum of the stiffnesses of the different parts of the screw; head, nut, threaded and unthreaded shafts. For a rod with cross sectional area A and length L, the axial stiffness becomes [13]

K = EA

L , (1.3)

where E is the Young’s modulus of the material. Hence, this method is independent of friction in the screw joint. The difficulties therefore lie in estimating the elongation as accurately as possible. A common approach is to measure the time of flight (TOF) for an ultrasonic pulse propagating back and forth through the screw, and thereafter determining the elongation from the difference in TOF (∆T OF ) between the untightened and tightened screw [3, 10, 14, 15]. The increase in TOF will partly be from the increased propagation path (by 2δ), but also from a decrease of ultrasonic wave speed due to the stress in the screw. This phenomena is known as the acoustoelastic effect and will be derived in Section 2.4. Further, a detailed explanation of how to determine the TOF, and thereafter calculate the preload will be given in Section 2.5.

Quite recently, studies have used a bi-wave pulse-echo technique to determine the preload in bolts. The technique is based on measuring the ratio of TOF between the longitudinal and transversal wave, and is therefore not in need of a calibration measure-ment of the untightened case [16, 17]. This technique will however not be investigated in this work.

In Section 2.6 a compressed sensing technique estimating the impulse response of the system is presented. The impulse response gives information about the secondary prop-agation paths that occur in the screw.

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1.4. Objective 5

1.4

Objective

The objective of this thesis was to analyse ultrasonic wave propagation in pre-stressed screws. More specifically the wavefront distortion was investigated, which physical phe-nomena it arises from and how it could affect ultrasonic measurements. Further, the possibility of describing the recorded signal as superpositions of multiple echoes, occur-ring from secondary propagation paths in the screw, was examined.

The purpose was to contribute to the research and development of measuring preload using ultrasonic measurements, increasing precision and accuracy of Atlas Copco’s tools. The experiments were conducted in a simulation environment using finite element anal-ysis (FEA). Numerical tools have then been used to process the data.

1.5

Limitations

The simulations were limited to temperature independence, elastic deformations and 2D axisymmetric behaviour. The analysis was limited to a certain number of degrees of freedom (element size), as well as the time step interval (sample rate). These limitations were chosen in order to reduce the computing power and complexity of the problem.

1.6

Outline

This thesis follows the structure of a scientific paper; Theory, Method, Results, Discussion & Conclusion. The two most important sections to get a grasp of are the Wavefront Distortion 2.3.5 and Compressed Sensing 2.6, which most of the results and discussion in this work are based on.

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C

HAPTER

2

Theory

This chapter will give the reader the necessary theory used throughout this thesis. The chapter is divided into several sections, where the first two sections go through solid mechanics and the finite element method (FEM). Next, some fundamentals of ultrasonic wave propagation and transducers will be given. The important acoustoelastic effect will then be derived, followed by ultrasonic preload measurements. Finally, an estimation of the impulse response using compressed sensing will be described.

2.1

Solid Mechanics

This section will go through some basic solid mechanics followed by some principles of elastic deformations. A deformation of a body can be caused by contact forces, body forces, or temperature changes within the material. When tightening a screw, the preload will act as contact forces at the head and threads in contact with the nut. A deformation beyond the von Mises yield criterion would transfer the material into the plastic region, meaning that the initial configuration cannot be reconfigured by removing the stress field [18]. All theory in this thesis are based on deformations in the elastic region. Further the isothermal assumption is made, meaning that the effect of temperature will not be taken into account.

2.1.1

Hooke’s Law

The Cauchy strain, ε, is defined as the ratio between total elongation, δ, to initial length, L, as

ε = δ

L. (2.1)

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8 Theory

For an isotropic linear elastic material with Young’s modulus E, the resulting strain, ε, due to a tensile load, σ, is found from Hooke’s law

ε = σ

E, (2.2)

which in tensor form states [19] εij = 1 2µ  σij − ν 1 + νσkkδij  , (2.3)

where δij is defined as the Kronecker delta, and σij is the Cauchy stress tensor

σij = 2νεij + λεkkδij, (2.4)

where λ and µ are the Lam´e constants given from Young’s modulus E, and Poission’s ratio ν as µ = E 2(1 + ν), λ = Eν (1 + ν)(1 − 2ν). (2.5)

2.1.2

Elastic Deformations

In continuum mechanics the material body is treated as a continuum of material points, an assumption made from the discrete points of atoms which the material consists of. Consider a point P in the specimen with position vector X relative to the origin O. If the material is deformed the displacement of point P can be described as

u = X − x, (2.6)

where x is the position vector to the same material point, denoted p. This is depicted in Fig. 2.1. A mapping function, χ, can then represent the deformation by taking all points X to x, such that

x = χ(X, t). (2.7)

Thereby, the mapping function, χ, mathematically describes the displacement field (some-times referred to as the strain field) for all points in the material. The displacement vector u(X, t) in Eq. (2.6) can be used to describe the true strain of a material point P as [19]

εij = 1 2 h∂ui ∂xj +∂uj ∂xi i , (2.8)

where i, j ∈ {1, 2, 3} denotes the cartesian coordinates. By taking the time derivatives of Eq. (2.7) the velocity and acceleration of a material point can be defined as

˙ x ≡ ˙χ = d dtχ(X, t), (2.9) ¨ x ≡ ¨χ = d 2 dt2χ(X, t). (2.10)

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2.1. Solid Mechanics 9

Further, a local deformation between two points in a material can be described as [20]

F = ∇Xx = ∇Xχ(X), (2.11)

where F is defined as a second order deformation gradient tensor with components Fij = ∂xi

∂Xj, and i, j ∈ {1, 2, 3} denotes the cartesian coordinates. The Jacobian is defined as

J ≡ detF > 0, with its important property of measuring the change in material volume. For the case without any rotation the Jacobian becomes [21]

J = ε1ε2ε3. (2.12)

The nominal stress tensor S can now be defined as

S = J F−1σ, (2.13)

which is a convenient expression, since it relates the forces in the current configuration with surface areas of the reference configuration [20].

𝑝 𝑃 𝑿 𝒙 𝑶 𝒙 = 𝜒(𝑿, 𝑡)

Figure 2.1: An illustration of the mapping function χ(X, t) during a deformation of a body. The position of the initial particle positions are located at vector X, and the position of the particles after deformation are located with the vector x.

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10 Theory

2.2

Finite Element Method (FEM)

In this section some brief fundamentals of finite element analysis (FEA) will be given. Further, the theory necessary to perform a pre-stressed transient analysis will be given for two different methods: mode superposition (MSUP) and the full method. For a further background in finite element analysis the reader is referred to Sunnansj¨o [22].

The finite element method (FEM) is used to find an approximate solution to a math-ematical model. From numerical treatment the model is discretized into a finite number of parameters, N , called degrees of freedom (DOF) [23]. There are numerous of different styles of the finite elements. For a planar model, the quadrilateral and triangular elements are used, which can be converted into axisymmetric elements. The point of connection between elements are defined as nodes. The relation between force and displacement acting on each node in a static system is given by [22]

Ku = F, (2.14)

where K is the global stiffness matrix, u the vector with nodal displacements, and F the vector with nodal acting forces. In general, FEM over estimates the actual stiffness of the model, which implies an underestimation of displacements. The style of the elements can also affect the model stiffness, where e.g. the use of triangular elements increase the stiffness of the model compared to rectangular elements. The solution will in general be more accurate in high stress regions, except for regions where high gradients occur [22]. The analysis can be assumed to be linear or nonlinear, where the computational time is reduced significantly with the linear model. A linear model means that the stiffness matrix K is assumed to be constant. This is valid if no large deformations occur dur-ing the solution. Otherwise the model should be solved with a nonlinear analysis, i.e. the stiffness matrix K will be calculated for every iteration during the solution. For a dynamical system the discretized equation of motion becomes

M¨u + C ˙u + Ku = F(t), (2.15)

where M is mass, and C damping coefficients.

2.2.1

Mode Superposition (MSUP) Method

The concept of the MSUP method is to reduce computational time needed by finding the natural frequencies and mode shapes of the system with a modal analysis. The DOF of the system is therefore reduced to the amount of mode shapes used.

Assume that the nodal displacements u can be written as a linear combination of mode shapes (eigenvectors) φi, with N DOF as

u = N X

i=1

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2.3. Ultrasonic Wave Propagation 11

where yi(t) are constants, describing the modal amplitudes. The equations of motion reduces to [24]

¨

yi+ 2ωiξiy˙i+ ωi2yi = Fi(t), (2.17) where ωi are the natural frequencies, and ξi the corresponding damping coefficients. For this method to perform properly it requires that the behaviour of the system can be assumed to be linear.

2.2.2

Full Method

The full method is more computational expensive since it solves the full equation of motion (2.15) step by step. The benefit by using the full method is that nonlinearities in the model are allowed. Previous studies have shown that to obtain good results the mesh size should be chosen such that the minimum wavelength contains at least ten elements [25, 26] ∆x ≤ λmin 10 = cl 10fmax , (2.18)

where ∆x is the elements size, cl the longitudinal wave speed (which is the maximum), and fmax the frequency of the sound wave. The maximum time step is then chosen such that the maximum wave speed is not able to cross one element in one time step

∆t ≤ ∆x cl

= 1

10fmax

. (2.19)

2.3

Ultrasonic Wave Propagation

An ultrasonic wave is defined as acoustic sound waves with frequencies greater than 20 kHz (the limit of human hearing). An ultrasonic wave propagates by particle displace-ments in the medium which it is propagating through (will later be treated as small deformations when defining the acoustoelastic effect in Section 2.4). The application for ultrasonic devices are numerous and can today be found in many different fields; medicine, non-destructive testing (NDT), flow measurements and more [27]. This section will give some basic theory about ultrasonic wave propagation.

2.3.1

Attenuation

As a sound wave propagates it will decrease in energy due to absorption and scattering in the material. Absorption is the conversion of energy from the sound wave to heat in the material, and scattering is a reflection in the material resulting from the fact that the material is not purely homogeneous. Attenuation is defined as the combined effect from these two. The remaining sound pressure after propagating a distance z is found with Beer Lambert’s law

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12 Theory

where p0 is the initial sound pressure, and α(ω) the frequency dependent attenuation constant [28].

2.3.2

Wave Modes

Basically there exists two different wave modes when propagating an ultrasonic wave in a solid; longitudinal (compression) and transversal (shear) waves. For a longitudinal wave, the motion of particles is in the same direction as the wave direction. This causes the particles in the medium to get compressed, hence longitudinal waves are sometimes called compression waves. For a transversal wave, the motion of particles is perpendicular to the direction of the wave. These two wave modes are depicted in Fig. 2.2. Longitudinal waves exists in gases, liquids and solids, while transversal waves only propagates significantly in solids (except from highly viscous liquids) [29]. Other wave modes such as surface and plate waves can also occur, for instance at the interface of two materials. However these wave modes will not be considered in this work.

Transversal wave Longitudinal wave Propagation direction Motion of particles Motion of particles Wavelength

Figure 2.2: Depiction of longitudinal and transversal wave propagation.

2.3.3

Reflection and Mode Conversion

When a sound wave hits the boundary between two materials it will refract and reflect due to a difference in impedance given from Snell’s law

c1 sin θ1

= c2 sin θ2

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2.3. Ultrasonic Wave Propagation 13

where c1and c2 are the sound velocities, and θ1and θ2the incident and reflected/refracted angles. This relation also holds for the mode-converted waves. The reflection coefficient, describing the sound pressure distribution after a reflection between two materials with acoustic impedance Z1 and Z2 is [28]

R = Z2− Z1 Z1+ Z2

, (2.22)

where Zi = ρici is the acoustic impedance of the material. The ratio between reflected and incident energy is by definition the square of Eq. (2.22). For a steel air interface, the reflected energy becomes R2 = 97%. Hence, almost all energy of the wave are converted into the reflected wave. Moreover, during a reflection the wave will undergo a mode conversion; such that all energy will not be kept in the longitudinal wave (which often is the wave mode of interest). The energy distribution of a reflected longitudinal and shear wave from an incident longitudinal wave is given by [30]

R2ll= "

cos22θ2− (cs/cl)2sin 2θ1sin 2θ2 cos2θ 2+ (cs/cl)2sin 2θ1sin 2θ2 #2 , (2.23) R2sl = 4(cs/cl) 2cos2 2sin 2θ1cos 2θ2 h cos2θ 2+ (cs/cl)2sin 2θ1sin 2θ2 i2. (2.24)

The corresponding distributions for an incident longitudinal wave propagating in steel are plotted in Fig. 2.3 as a function of incident angle. Note that the reflected shear wave will contain more energy than the longitudinal wave for incident angles between 28 and 87 degrees. 0 10 20 30 40 50 60 70 80 90 0 0.2 0.4 0.6 0.8 1

Figure 2.3: Energy distribution as a function of incident angle for the reflected shear and longitudinal wave in steel, with cl = 5900 m/s, and cs = 3250 m/s.

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14 Theory

2.3.4

Piezoelectric Transducers

The function of an ultrasonic piezoelectric transducer is to convert an electrical signal into mechanical vibrations, and then convert the received vibrations back into electrical signals. Piezoelectric transducers are doing this by applying an electric field across a polarized material, which causes the molecules to align and change the dimension of the material, generating a mechanical pulse [31]. The electro-mechanical coupling factor is an indicator of the conversion from electric voltage to mechanical displacement or vice versa. This value is typically between 10-20% [28]. There are numerous of different transducers available, where they differ in frequency, bandwidth and signal design. In this work the transducers are assumed to be circular with the property to transmit a given signal as a function of time and amplitude. Further, the transmitted ultrasonic pulse will spread due to the fact that the particle vibrations does not always transfer all of its energy in the wave propagation direction, simply because the particles in the medium are not perfectly aligned. This is known as beam divergence, which can be reduced by increasing the transducer diameter and operating frequency [27].

2.3.5

Wavefront Distortion

The wavefront of the transmitted pulse can be described from the characteristics of the ultrasonic transducer, however the wavefront at initial time will be assumed to be planar. Further the recorded echo after propagating back and forth through the screw will not have the same wavefront shape; it will get distorted. Distortion of the wavefront occurs from several physical phenomena, where the impact on the results depends on the system and transmitted signal. In this work the wavefront distortion will be considered to arise from three different phenomena; dispersion, secondary propagation paths, and non-homogeneous stress fields. If the end of the screw is not flat it will also cause a distortion of the wavefront. Screws with a flat end have therefore been considered in this work.

Dispersion

Dispersion is a phenomena which occurs when the phase velocity is frequency depen-dent, i.e. sound waves with different frequencies will propagate with different velocities. Dispersion occurs for ultrasonic propagation through both tightened and untightened screws. The recorded signal will be an integrated result over the transmitted frequencies, which will appear as a wavefront distortion. This also complicates the definition of the TOF, since it will be a function of frequency.

Secondary Paths

Due to divergence the beam will spread and can not be considered to follow a straight line along the screw. The shortest path will be sound waves that only have reflected at the

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2.3. Ultrasonic Wave Propagation 15 (a) (b) (c) 𝜃 L L L S (d)

Figure 2.4: Secondary echoes due to sidewall reflections. In (a): rhombic path, (b): two different rhombic paths, (c): echoes resulting from small offsets, θ, from the middle line, and in (d): mode-converted echo, where L = longitudinal wave and S = shear wave.

end surface of the screw and then returned back to the transducer. However, a countless number of other paths will also occur (some presented in Fig. 2.4), and depending on the geometry of the screw the time delay might be short enough to interfere with the shortest path echo. Additionally, for each reflection with the boundary of the screw the pulse experiences a phase shift of π radians [28]. In Fig. 2.4a a rhombic path in a cylinder is shown, where the time delay according to the shortest path is

τ1 =

2√D2+ L2 − 2L cl

, (2.25)

where L and D is the length and diameter of the screw. For an M8 screw with length 70 mm and transmitting a 5 MHz pulse, the delay of the rhombic path is approximately one wavelength of the pulse. The two paths shown in Fig. 2.4b will have time delay

τ2 = 4 3 √ 9D2+ 4L2− 2L cl , and τ3 = 3√4D2+ L2 − 2L cl , (2.26)

which for the most scenarios arrives a couple of cycles after the shortest path echo. Another path to consider is the one shown in Fig. 2.4c, which reflects at the end of the screw and then once more before returning to the transducer. The last path that has been considered in this work is a mode-converted echo, drawn in Fig. 2.4d. The delay of this path is therefore dependent on both the longitudinal and transversal wave speed as

τ4 = D cl s c2 l c2 s − 1. (2.27)

This echo arrives somewhere between the shortest path and the paths shown in Fig. 2.4b. The derivation leading to (2.27) can be found in Appendix B. As mentioned there will be a countless number of paths, several slightly shorter or longer than those presented

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16 Theory

above. The delay of these secondary echoes will also depend on the acoustoelastic effect, which therefore creates the possibility that they also could be used to determine the preload.

Wavefront Curvature

For a tightened screw, heterogeneities in the material will occur near the head and nut. More specifically the stress will be greater near the boundary of the screw, causing the ultrasonic waves to propagate with a decreased velocity further from the center of the screw. Thus, the wavefront gets distorted (see Fig. 2.5), which could cause difficulties when analysing the received echo. This problem also occurs in medical imaging where heterogenous tissue layers causes distortion of the wavefront. This has been studied by many groups, where for instance Huang and Tsao have presented a correction algorithm using the angular spectrum method and backpropagation of the received signal [32].

Propagation direction Wavefront y [mm] x [m m ] x [m m ]

Figure 2.5: A theoretical wavefront distortion when only considering the impact from a heterogeneous stress field. The estimated stress field is shown in the upper plot and the expected wavefront curvature (excessively scaled) is shown in the lower plot. The wavefront will stay consistent where the stress is homogenous along the x-axis.

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2.4. Acoustoelastic Effect 17

2.4

Acoustoelastic Effect

The acoustoelastic effect is a non-linear effect where the sound velocity of a material changes with an applied stress. This phenomena was first discovered in 1925 by Bril-louin [33]. A correct mathematical theory was later presented by Murnaghan in 1937 [34]. This theory included three third-order elastic constants l, m, and n, which was finally confirmed in 1953 by Hughes and Kelly [35].

Expressions for the longitudinal, cl, and shear, cs, wave velocities have been derived from the generalised Hooke’s law and Cauchy’s stress equation for the case of an elastic isotropic material. The sound velocities for an unstressed material are [36]

cl = s λ + 2µ ρ , cs= r µ ρ, (2.28)

where ρ is the density of the material, and λ and µ are the second order Lam´e-constants given in Eq. (2.5). Conversely the velocities in Eq. (2.28) can be written as

cl = s E(1 − ν) ρ(1 + ν)(1 − 2ν), cs= s E 2ρ(1 + ν). (2.29)

As mentioned the sound velocities will change if the screw is elongated, which will be determined in the next section.

2.4.1

Small-on-large Theory

The sound propagation in the material can be seen as a small local deformation. Con-sidering that the material experience this smaller deformation in addition to the larger deformation caused by the applied stress is known as the small-on-large theory [21]. Using Eq. (2.6) describing the deformation yields the total displacement as

u = u(0)+ u(1) (2.30)

where u(0) = x − X is the large deformation, and u(1) = x0− x is the small deformation, such that

u = x0− X. (2.31)

The mapping function becomes

χ(X, t) = u + X. (2.32)

The small-on-large theory is based on the assumption [37]

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18 Theory

which indeed holds for this work where the deformations caused by the sound waves are much smaller than the deformations caused by the preload. The standard approach using small-on-large theory is to perform a Taylor expansion of the equation of motion, which makes it possible to determine physical parameters, such as wave speeds as a function of preload [37]. Cauchy’s first law of motion states

∇S = ρ¨χ, (2.34)

and with insertion of Eq. (2.32), the right hand side becomes ρ¨χ(X, t) = ρ∂ 2 ∂t2  u(0)+ u(1)+ X= ρ∂ 2u(1) ∂t2 , (2.35)

with the assumption that the initial and unstressed configurations are static, i.e. the time derivatives are zero. The derivation of the left hand side of Eq. (2.34) will be described very briefly, where the reader is referred to Eldevik for a more detailed derivation [20]. First, the displacement gradient is introduced as

∇u = F − I, (2.36)

where F was defined in Eq. (2.11), and I is the identity matrix. Further a Taylor expan-sion of the strain energy is performed, which is used to rewrite the nominal stress tensor in Eq. (2.13). The gradient ∇S is expanded with the chain rule of the displacement derivatives and the assumption that the initial deformations are in equilibrium. Finally, Eq. (2.34) becomes Bijkl ∂2u(1) k ∂xj∂xl = ρ∂ 2u(1) i ∂t2 , (2.37)

where Bijkl is a constant dependent on the displacement vectors u, and the second and third order elastic moduli [21]. Note that Eq. (2.37) has the form of a linear wave equation.

2.4.2

The Plane Wave

Consider a plane wave propagating in the direction of the unit vector n (as in Fig. 2.6)

u(1)(x, t) = mf (n · x − ct), (2.38)

where c is the phase velocity, and f is a twice continuously differentiable function [20]. The polarisation m satisfies from Eq. (2.37)

ρc2m = Λm, Λik ≡ Bijklnjnl, (2.39) where ρc2 is the eigenvalue of the acoustic tensor Λ for the eigenvector m [37]. For a longitudinal wave n = m, and for a transverse wave n ⊥ m. The wave speeds are then determined by the characteristic equation

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2.4. Acoustoelastic Effect 19 n m ct Wavefront at t = 0 Wavefront at time t

Figure 2.6: Depiction of a plane wave propagating in the normal direction n with polarisation m.

Now, consider a tightened screw. The longitudinal and transverse wave speeds be-comes [38] cii(z) = s λ + 2µ + aiiε(z) ρ , (2.41) cij(z) = s µ + aijε(z) ρ , (2.42)

where aii and aij are the acoustoelastic coefficients related to effects from third order elastic constants. The coefficients for propagation in the direction of tension are

a11 = µ λ + µ(λ + 2B + 2C) + 2(2λ + 5µ + A + 2B), (2.43) a12 = µ λ + µ h 4(λ + µ) +λ + 2µ 4µ A + B i , (2.44)

and the coefficients for propagation perpendicular to the direction of tension are a22 = − 2λ(λ + 2µ) + λA + 2(λ − µ)B − 2µC λ + µ , (2.45) a21 = (λ + 2µ)(4µ + A) + 4µB 4(λ + µ) , (2.46)

where the constants A, B and C are third order elastic constants for an isotropic hy-perelastic material ascribed by Landau & Lifshitz in 1986 [39]. These are related to the Murnaghan constants as

l = B + C, m = 1

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20 Theory

Consequently the sound velocities can be found from the stress, σ(z), instead of pre-strain, ε(z), by replacing ε(z) in Eq. (2.41)-(2.42) with the Cauchy strain presented in Eq. (2.3) ε(z) = σ(z) E = λ + µ 3κµ σ(z), (2.48) where κ = λ +2 3µ (2.49)

is known as the bulk modulus. Further, Eq. (2.41) can be linearized in the first order as [40]

c11(z) = c (0)

l 1 + A11(z)σ(z), (2.50)

where c(0)l is the longitudinal velocity for the untightened screw given by Eq. (2.28), and A11(z) is known as the acoustoelastic constant.

2.5

Ultrasonic Preload Measurements

The principle of ultrasonic preload measurements is to estimate the TOF for an ultrasonic pulse propagating back and forth through the screw, and then calculate the difference in TOF, ∆T OF , between the unstressed and preloaded state. Assuming that the material properties are known, the preload can be determined with a small margin of error if ∆T OF is estimated accurately.

Recall from Section 1.3.2 that preload can be found from the product of the total axial stiffness and elongation of the screw: Fpre = Kδ (1.2). If one could ignore the influence of the acoustoelastic effect, the elongation would simply be obtained from ∆T OF as

δ = ∆T OF

2 cl, (2.51)

but this is not true since the sound velocity will change due to an applied stress. In fact, the contribution to an increase of TOF is roughly 1/3 from geometrical elongation, and 2/3 from the acoustoelastic effect. The TOF for the tightened screw should be expressed as T OF = 2 Z L+δ 0 1 + ε(z) c11(z) dz, (2.52)

where the assumption that the stress is homogeneous along a cross section of the screw is made. The stress dependent longitudinal sound velocity c11(z) is given from Eq. (2.41). Assume that the TOF is measured for both a tightened and untightened screw. The total elongation, which gives the preload, can be determined by dividing the integral expression in Eq. (2.52) into a summation over the different parts of the screw with different stiffness (i.e. head, shank and threaded part) as in Eq. (1.3).

What remains is to estimate the TOF for the two cases; untightened and tightened screw. The TOF can simply be estimated by finding the maximum peak of the received

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2.6. Compressed Sensing 21

echo, which however is not a reliable method. Instead the cross-correlation between the transmitted and received signal could be performed, and finding the maximum peak of the correlation spectrum. The most promising method has been to model the signal with a Barker code [41]. A Barker code is a binary code modeled such that the autocorrelation spectrum has a maximum sidelobe of 1. The longest known Barker sequence has a length of 13 bits [42]

1 1 1 1 1 0 0 1 1 0 1 0 1, (2.53)

with its autocorrelation sequence plotted in Fig. 2.7. This sequence can be modeled onto an ultrasonic signal by introducing phase shifts of π radians.

Estimating the preload from TOF suffers from inaccurate measurements, sample rate and wavefront distortion. In the next section a parametric model of the system will be derived, which instead exploits the fact that the wavefront gets distorted.

-10 -5 0 5 10

0 5 10

Figure 2.7: Autocorrelation of a Barker-13 sequence, Eq. (2.53).

2.6

Compressed Sensing

The purpose of an empirical model is to find a model of the system based on its physical properties, where it is important to select a model structure that represents the true system. The model presented below will be based on the assumption that the recorded signal can be written as superpositions of delayed and scaled versions of the transmitted pulse.

Each component in the system has a transfer function; electrical, transducer, the screw, and the couplant between transducer and screw head, where the transducer typically has the narrowest bandwidth, dictating the frequency spectrum of the transmitted signal [43]. A good technique is therefore to sample the signal such that both the first and second echo are recorded. The first echo is then used as an input and the second echo as an output,

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22 Theory

concluding that the transfer function between them is only the system containing the screw properties. During the simulations the input and output signals will be compared where the output signal only has been affected by the transfer function subjected to the screw.

The wave equation in one space dimension (displacement is assumed to only occur in one direction) is [4]

∂2P (x, ω) ∂x2 = −k

2

(ω)P (x, ω), (2.54)

where P (x, ω) is the acoustic pressure and k(ω) the complex frequency dependent wave number

k(ω) = ω cp(ω)

− jα(ω), (2.55)

including the phase velocity, cp(ω), and attenuation constant, α(ω) ≈ αω2. The solution to the ODE in (2.54) is

P (x, ω) = P (0, ω)e−jk(ω)x. (2.56)

A non-parametric estimation can for instance be obtained with the Fourier transform ˆ

H(ω) = Y (ω)

U (ω), (2.57)

where Y (ω) is the recorded acoustic pressure after propagating a distance x, and U (ω) the transmitted signal at x = 0. The transfer function for a propagation path q can be modeled as

Hq(x, ω) = e−jωτq(x,ω)−αω

2x

, (2.58)

where τq = x/cp(ω) is the time delay for path q. The total transfer function of the system is assumed to be the sum of Q paths. By assuming that the phase velocity and attenuation constant are frequency independent the inverse Fourier transform gives

h(t) = Q X

q=1

αqδ(t − τq). (2.59)

The received signal is thereby modeled as the superposition of reflected echoes

y(t) = u(t) ∗ h(t) + e(t) = Q X

q=1

αqu(t − τq) + e(t), (2.60)

where e(t) is some random noise. The remaining task is to find a good estimate of the transfer function ˆh(t). The true model in (2.59) is in continuous time, but will be discretized into a number of samples, N , depending on the sampling rate and time. Eq. (2.60) in matrix form becomes [6]

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2.6. Compressed Sensing 23

where h and y are vectors and U the convolution matrix designed from u. h can now be recovered from measured data with for instance the least-squares solution [44]

hLS = arg min h ||Uh − y||2 2 = (U TU)−1 UTy, (2.62)

which returns the optimal solution in least-squares sense. The solution is however often unstable due to rank deficiencies in U and measurement noise. Thus a regularization parameter µrr is often added as

hRR = arg min h

||Uh − y||22+ µrr||h||22 = (U T

U + µrrI)−1UTy, (2.63)

where I is the unit identity matrix. This method is called ridge regression (RR), also known as Tikhonov regularization. When µrr = 0 the expression becomes the ordinary least squares, but when µrr> 0 the addition of µrrI ensures that the matrix is invertible and a stable solution is obtained. Further if the system is known to be sparse, i.e. the number of non-zero elements in h is small, a least absolute shrinkage and selection operator (LASSO) is a good modification of the problem, such that [45]

hCS = arg min h

||Uh − y||22+ γ||h||1, (2.64)

where instead of penalizing with the `2 norm as in (2.63), the penalty function is the `1 norm. CS denotes compressed sensing, which basically is the engineer’s terminology of LASSO. Penalizing with the `1 norm will force more entries of h to equal zero, i.e. the technique yields sparse models that are more easily interpreted. Previous studies have shown that this technique outperforms `2 regularization, when the model is known to be sparse and/or irrelevant features are present in the measured data [6, 46, 47, 48]. Unlike least-squares and Tikhonov regularization, Eq. (2.64) does not have a closed form solution, instead the problem is generally solved with quadratic programming and convex optimization [48, 49].

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3

Method

This chapter will thoroughly describe the numerical tools that have been used extensively throughout this thesis. Moreover the simulation models will be described step by step followed by the post-processing methodology.

3.1

Finite Element Analysis

The finite element analysis (FEA) in this work have been performed using the simulation and 3D design software ANSYS 2019 R1. The simulations have been performed in ANSYS Mechanical using analysis type 2D and axisymmetrical behaviour, i.e. the models were symmetric around the y-axis. Thus the planar elements were converted into axisymmetric elements. The mesh element size was set to 0.1 mm, which corresponds to approximately 11 elements per wavelength for the 5 MHz pulse that has been used. Thus Eq. (2.18) is fulfilled. The geometry was modeled in SpaceClaim, a 3D modeling CAD software. In Fig. 3.1 a modeled M8 screw can be seen. The thread dimensions are given from the ISO standards (see Appendix A). The material properties for structural steel are presented in Tab. 3.1.

x y

Figure 3.1: A CAD-model of a 2D axisymmetric M8 screw created in SpaceClaim.

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26 Method

Table 3.1: Material properties; density (ρ), Young’s modulus (E) and Poisson’s ra-tio (ν) of structural steel implemented in ANSYS Mechanical. The last three columns shows the Murnaghan constants for steel (in GPa), which was used for analytical post-processing [50].

Density (ρ) Young’s modulus (E) Poisson’s ratio (ν) l m n

7850 kg/m3 200 GPa 0.3 -86 -629 -728

3.1.1

Pre-stress

The pre-stressed model was accomplished by applying a fixed support at the head of the screw along with seven forces distributed at the threads in contact with the nut. The load distribution applied on these threads is given in Fig. 3.2 [51].

3.1.2

Transient Analysis

A transient analysis was run for a couple of cylinders and screws with various dimensions, for instance the M8 presented in Fig. 3.1. The method used to solve the equations of motion (2.15) implemented in ANSYS were the Newmark time integration method, or an improved method called HHT [24]. The element size and time step were the two most crucial choices for the transient analysis since they both had a huge impact on the computational time.

Pulse Generation

The ultrasonic pulse was generated by either applying a time-varying force or displace-ment on the head of the screw where the transducer was connected. The radius of the cylindrical ultrasonic transducer was set to 2 mm. The reason to this was to minimize the effect from divergence of the beam, which decreases with an increased transducer area. A

34% 23% 15% 11% 7% 5% 3% 1 2 3 4 5 6 7 0 10 20 30 40 Load [%]

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3.1. Finite Element Analysis 27

high divergence creates a lot of reverberations from the threads, which interferes with the reflection from the end of the screw. For the simulations without preload, displacement as a function of time was used. According to Krautkr¨amer an applied effect of 10 W/cm2 would in steel translate into a maximum displacement of 1.8 · 10−6 wavelengths, which for a wavelength of 1 mm becomes 1.8 nm [28]. An amplitude of 1 nm was used for the excited pulse. A common pulse used when performing ultrasonic testing is the Gauss filtered cosine. It is convenient for post-processing since the product and convolution of two Gaussian functions is still Gaussian. It is also Gaussian in the frequency domain, with a known spectral 1/e width equal to 1/2σ. The pulses used during the simulations have the following expression

u(t) = A sin(2πωt)e−(t−µ)22σ2 , (3.1)

which for arbitrarily values of ω, µ and σ are plotted in Fig. 3.3. The received echoes were then recorded with a directional deformation probe placed on the same area. The Barker modulated pulse was generated by using tabular data as look-up values for the displacements. 1 1.5 2 2.5 3 10-6 -5 0 5 10-10 -5 0 5 106 2 4 6 8 10 12 10-8

Figure 3.3: Left: Gaussian pulse generated from Eq. (3.1), with ω = 5 M Hz, σ = 0.25 · 10−6, µ = 2 µs, and A = 1 nm. Right: Magnitude of corresponding frequency domain, analytically determined with a Fourier transform.

MSUP

This method was performed by first finding the desired amount of modes from a modal solution with the pre-stress computed from a static structural model. The time step for the MSUP method was then set from the following relation [24]

∆t < 1 20fresponse

, (3.2)

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28 Method

Full Method

The analysis settings for the full method are divided into three steps. The first step applies the preload (if any) as described above in 3.1.1, with time integration turned off. The second step transmits the ultrasonic pulse, and then the last step propagates the pulse as well as samples the displacement at the transducer area. The maximum time step was chosen such that the maximum wave speed did not cross more than one element in one time step, see Eq. (2.19).

3.1.3

Pre-stressed Transient Analysis

A biased pre-stressed transient analysis was created around an operating point of 50 kN. The number of significant digits was set to 10 (which was the maximum), because the displacements occurring from the US waves were very small compared to the strain field resulting from the preload. The numerical damping value was set to 0, such that the US waves were not perceived as noise. Since the nodes connected to the transducer will be displaced after the first step declaring the preload, a time-varying force (50 N) was used to excite the US pulse. The option ”Large deflections” has been used, i.e. the stiffness matrix was calculated for each time step in order to correctly account for the elongation of the screw. To account for the acoustoelastic effect, an orthotropic material was defined, i.e. Young’s modulus (E), Poisson’s ratio (ν) and the shear modulus (µ) were defined for x, y, z, xy, xz, and yz-directions. These constants were calculated for a preload of 50 kN, and thus introduced a bias around that load. The constants in x, y, and z-direction were derived from the acoustoelastic velocities (2.41-2.42), and the cross terms were taken as the mean between them. The resulting constants are presented in Tab. 3.2. The material became homogeneous in the sense that the sound velocities were constant for the whole screw. Thus the wavefront curvature did not arise during these simulations. The theoretical elongation, δ, was obtained from Eqs. (1.2)-(2.4), which can be found from a given stress, σ, or preload, Fpre, as

δ = σLclamp

E =

FpreLclamp

EA , (3.3)

where the clamp length, Lclamp, was taken from the head of the screw to the first thread where the nut was applied. Since the model was such that the axial stress was equal in

Table 3.2: The material constants used to define the orthotropic elastic material around the operating point 50 kN. Young’s modulus (E) in GPa, Poisson’s ratio (ν), and the shear modulus (µ) in GPa are defined for different directions.

Ex Ey Ez νxy νyz νxz µxy µyz µxz

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3.2. Post-processing 29

all points, the theoretical ∆T OF was calculated as (2.52) ∆T OF = 2(L + δ)

c11

, (3.4)

where the longitudinal wave speed, c11, was calculated from Eq. (2.41).

3.2

Post-processing

The post-processing of the resulting data have been performed in the numerical comput-ing environment MATLAB R2019a. The sampled displacements at the transducer have been stored as column vectors. Since the transient analysis uses auto time stepping the resulting data will not have constant time steps. The data were therefore interpolated such that the time step became constant for all samples. The normal stress have been exported as text files containing the nodal positions and the stress in x- and y-direction for each node.

3.2.1

Wavefront Curvature

A numerical investigation of the wavefront curvature has been performed by importing the text files with stress and nodal positions in MATLAB. The stress field was then converted into local velocities from the acoustoelastic effect (2.41). Thus only longitudinal waves and a propagation only in y-direction were considered, such that the wavefront could be divided into propagation paths following a straight line. The width of the wavefront was set to the diameter of the US transducer (4 mm). The time delay after propagating back and forth through the screw was then calculated for each path.

3.2.2

Compressed Sensing

The models introduced in Section 2.6 have numerically been solved in MATLAB. Some white Gaussian noise, e(t), was added to the sampled displacements, y(t), with various signal-to-noise ratio (SNR). To solve the compressed sensing problem in Eq. (2.64) CVX was used, a package for specifying and solving convex programs [52, 53]. The package uses conic solvers, meaning that the problem to be solved should be constructed with norms, increasing speed and accuracy. Even though a problem could be stated such that the cardinality of h (number of non-zero elements = `0 norm) should be less than a certain number, it is way more efficient in the sense of optimization to find the parameter γ in (2.64) representing the same solution. Further, the problem in (2.64) can be recast and solved as a second-order cone programming (SOCP) problem, which with great efficiency can be handled by extensions of interior-point methods [49]. Lastly, a Tukey window was applied on the measured data, y, to prevent ill-conditioning at the boundaries.

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Results

This chapter will present the final results of this work. First, some wave propagation in ANSYS will be presented to get a visual overview of the simulation results. Next, the analytical wavefront curvature resulting from the heterogeneous radial stress field will be presented. Thereafter, the recorded signals at the transducer when performing transient analysis for cylinders and screws will be given. The results from the pre-stressed transient analysis model are then presented for an M8 screw. Finally, the estimated impulse response using a compressed sensing technique will be compared to LS and RR estimates for untightened and tightened screws.

4.1

Wave Propagation

All transient analysis presented in this chapter are performed using the full method, simply because the runtime of the MSUP method was longer. The four figures in Fig. 4.1 shows the von Mises stress at different time instances for a cylinder and an M8. The wavefronts taking the shortest path (straight line) are clearly visible in the figures. The reflected shear and longitudinal waves as well as the the mode-converted echoes can also be seen both for the cylinder and M8. The reverberations from the threads are also visible in Fig. 4.1c.

4.2

Secondary Echoes

This section will investigate the influence from secondary paths described in Section 2.3.5. The TOF presented in this section have been calculated by finding the peak of the cross-correlation between transmitted and received signal. No successful results using Barker modulated pulses were completed. It seemed like the discrete phase shifted pulse did not resolve properly with the element size and time step used. In Fig. 4.2 the recorded displacement after transmitting a Gaussian pulse is shown for a simulation using an M8

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32 Results

screw. The cross-correlation spectra is also shown, where three distinctive peaks can be seen; one with the transmitted signal, one from the shortest path echo, and one from the mode-converted echo. In Fig. 4.3 the recorded displacement when transmitting a Gaussian pulse in a cylinder is shown for three different diameters. The TOF for the first and second (mode-converted) echoes and the time delay between them are presented in Tab. 4.1. Note that as the diameter is increased both the TOF and amplitude of the first echo increases as well. Additionally, a simulation with the same set-up was run for

Reflected S-waves L-waves (mode-converted) Mode-converted waves Mode-converted echo

(c)

(a)

(b)

(d)

Figure 4.1: The figures show the equivalent von Mises stress at different time instances; (a): von Mises stress at time short after transmission, (b) and (d): von Mises stress at time when first wavefront reaches end surface, and (c): von Mises stress at time when the pulse reaches the receiver. Since the von Mises stress is symmetric around zero stress the colormap is scaled such that only positive values are shown, making the image less messy. The figures are also duplicated around the y-axis. In (a), (b) and (d) the 70 mm long cylinder with 8 mm in diameter has been used, and in (c) the 70 mm long M8 is used. The reflected shear (S) and longitudinal (L) waves are shown, as well as the mode-converted echo resulting from the path presented in Fig. 2.4d.

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4.2. Secondary Echoes 33 0 0.5 1 1.5 2 2.5 3 3.5 10-5 -5 0 5 10-7 0 0.5 1 1.5 2 2.5 3 3.5 10-5 0.2 0.4 0.6 0.8

Figure 4.2: Top: Sampled displacement at the transducer for a 70 mm long M8 screw (see Fig. 3.1). The material properties are given in Tab. 3.1. The transmitted pulse has the same specifications as in Fig. 3.3. Bottom: cross-correlation with the transmitted signal, used to estimate TOF.

a cylinder of length 100 mm and diameter 8 mm. The first echo arrived after 35.559 µs with a time delay of 2.134 µs to the mode-converted echo, which is close to the theoretical delay of 2.16 µs. A comparison between the recorded displacement for a cylinder and screw was made for a diameter of 8 and 10 mm in Fig. 4.4. The calculated TOF for the echoes can be found in Tab. 4.2. The TOF for the echo in the 8 mm cylinder corresponds to a longitudinal velocity of

cl= 2L

T OF = 5627.9 m/s, (4.1)

and for the M8

cl= 5648.3 m/s, (4.2)

this can be compared to the theoretical longitudinal velocity using Eq. (2.28) and the material properties in Tab. 3.1

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34 Results 0 0.5 1 1.5 2 2.5 3 3.5 10-5 -5 0 5 10-7 D = 8 mm D = 9 mm D = 10 mm 2.6 2.65 2.7 2.75 2.8 2.85 2.9 2.95 3 10-5 -5 0 5 10-7

Figure 4.3: Top: Sampled displacements in a 70 mm long cylinder for three different diameters; 8, 9 and 10 mm. Bottom: Zoom on the first two echoes.

Table 4.1: Calculated TOF from cross-correlation between transmitted and received sig-nal for the three different diameters of a cylinder, presented in Fig. 4.3. The fourth column shows the delay to the mode-converted echo, and the fifth column shows the the-oretical delay, obtained from Eq. (2.27) and Tab. 3.1.

Diameter First echo Mode-converted echo Time delay Theoretical

8 mm 24.872 µs 27.113 µs 2.241 µs 2.16 µs

9 mm 24.902 µs 27.318 µs 2.416 µs 2.43 µs

10 mm 24.922 µs 27.719 µs 2.797 µs 2.70 µs

4.3

Pre-stressed Transient Analysis

A preload of 50 kN was applied according to Section 3.1.1. The pre-stressed transient analysis was then performed as described in Section 3.1.3. The sampled displacements for a preload of 50 kN are compared with no preload in Fig. 4.5. The clamp length was measured to (see Fig. 4.6)

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4.3. Pre-stressed Transient Analysis 35 2.6 2.7 2.8 2.9 3 10-5 -5 0 5 10-7 M8 C8 2.6 2.7 2.8 2.9 3 10-5 -4 -2 0 2 4 10-7 M10 C10

Figure 4.4: Zoom on the first two echoes for a 70 mm long C8 and M8 at left, and C10 and M10 at right.

Table 4.2: Calculated TOF from cross-correlation between transmitted and received signal for the M8 and M10 screw presented in Fig. 4.4. The fourth column shows the delay between the echoes, and the fifth column shows the theoretical delay from Eq. (2.27) and Tab. 3.1. The TOF for the mode-converted echo in the M10 is undefined since there was no significant correlation peak.

Diameter First echo Mode-converted echo Time delay Theoretical

8 mm 24.786 µs 27.287 µs 2.501 µs 2.16 µs

10 mm 24.746 µs - - 2.70 µs

The theoretical elongation for a cylinder with diameter 8 mm and clamp length 57 mm exposed to a preload of 50 kN becomes

δ = 0.285 mm. (4.5)

However for an M8 screw the inner diameter (see Fig. A.1) is equal to Dmin = 7.46 mm, resulting in an elongation of

δ = 0.379 mm. (4.6)

The elongation obtained from the structural analysis in ANSYS was

δ = 0.384 mm. (4.7)

The theoretical ∆T OF using δ from (4.6) becomes

∆T OF = 427 ns. (4.8)

The ∆T OF calculated using cross-correlation from the displacements in Fig. 4.5 becomes

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36 Results 0.5 1 1.5 2 2.5 3 3.5 4 10-5 -2 0 2 10-6 0 kN 50 kN 2.7 2.75 2.8 2.85 2.9 2.95 3 3.05 3.1 3.15 10-5 -2 0 2 10-6

Figure 4.5: Top: Sampled displacement for an untightened and tightened M8 screw with 50 kN. The bottom figure shows an enlargement of the two first echoes.

4.4

Wavefront Curvature

The wavefront curvature was analytically determined for an M8. The stress field obtained when applying a preload of 50 kN and the estimated wavefront after propagating back and forth through the tightened screw are shown in Fig. 4.6. The wavefront is plotted with a relative position to the slowest part of the wavefront. The fastest part of the wavefront (occuring in the middle of the screw) is about 10 µm ahead of the slowest part. This would for a 5 MHz pulse correspond to a delay of 1/100 of a wavelength.

4.5

Compressed Sensing

Fig. 4.7 shows the estimated impulse responses using Eqs. (2.62)-(2.64) and the modeled reflections for a 70 mm long cylinder with 8 mm diameter. The time delay from the tap representing the transmitted pulse to the most significant tap representing the shortest path echo is

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4.5. Compressed Sensing 37 -1 0 1 0 2 4 6 8 10-6 Wavefront

Figure 4.6: Left: Normal stress in y-direction resulting from a preload of 50 kN dis-tributed over the last threads according to Fig. 3.2. The plot is duplicated around the y-axis for visual improvement. Right: Analytical estimation of the wavefront curvature after propagating back and forth through the screw. The relative positions are calculated using Eq. (2.41) with the material properties and Murnaghan constants for steel in Tab. 3.1.

and the TOF to the mode-converted echo is

T OF = 27.05 µs. (4.11)

Fig. 4.8 presents the impulse responses using a compressed sensing technique for the untightened and tightened M8 screw presented in Fig. 4.5. The TOF for the shortest path echo propagating in the untightened screw is

T OF = 24.63 µs, (4.12)

and for the tightened screw

T OF = 25.10 µs, (4.13)

taking the difference gives

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38 Results 0 1 2 3 10-5 -0.1 0 0.1 0.2 h LS (t) 0 1 2 3 10-5 -1 -0.5 0 0.5 1 0 1 2 3 10-5 -0.05 0 0.05 0.1 h RR (t) 0 1 2 3 10-5 0 0.2 0.4 0.6 0.8 1 h CS (t)

Figure 4.7: Top right: Simulated and modeled signal for a 70 mm long cylinder with 8 mm diameter. The modeled signal is modeled with the CS estimate shown in the bottom right figure. Left: LS and RR impulse response estimates. The recorded displacements are normalized. The measured signal was modeled with AWGN with SNR = 30 dB. The regularization parameters are µrr = 0.1 and γ = 0.5.

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4.5. Compressed Sensing 39 -0.5 0 0.5 1 h CS (t) 0 1 2 3 10-5 0 kN -0.5 0 0.5 1 h CS (t) 0 1 2 3 10-5 50 kN -0.5 0 0.5 1 h CS (t) 0 1 2 3 10-5 0 kN -0.5 0 0.5 1 h CS (t) 0 1 2 3 10-5 50 kN

Figure 4.8: Estimated impulse responses using compressed sensing (2.64) of the mea-sured displacements in Fig. 4.5. The two upper plots are generated using γ = 0.5, and the two bottom plots are generated with γ = 0.8. The SNR is set to 30 dB.

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C

HAPTER

5

Discussion & Conclusion

This final chapter will summarize and give some interpretations of the results. Some fu-ture improvements of the simulation model will also be highlighted. Finally, a conclusion followed by a couple of ideas on future work will complete this thesis.

5.1

Interpretation of Results

The measured longitudinal sound velocities (4.1)-(4.2) are slightly less than the theoret-ical speed (4.3). This might be a result from the fact that FEM in general overestimates the actual stiffness of the model [22]. This would imply an underestimate of displace-ments, and further also the sound velocities.

The figures presenting mode conversion in Fig. 4.1 and the time delays presented in Tab. 4.1 from the results in Fig. 4.3 indicate that the second significant echo is indeed a mode-converted echo. The mode-converted echo is lower in amplitude when simulating screws compared to cylinders. This makes sense since some of the energy reflects back due to the threads. The echo also appears to be protracted, probably because there is no straight sidewall. Moreover the echo seems to be the superposition of multiple mode-converted echoes reflected somewhere between the inner and major diameter of the screw.

It was expected that the first echo taking the shortest path should arrive indepen-dently of screw diameter. However in Fig. 4.3 the TOF of the first echo increases with an increased diameter. This would make sense if the echo is a superposition with reflec-tions taking secondary paths as described in Section 2.3.5. An increase of diameter also increases the propagation time of the echoes interfering with the shortest path echo.

When comparing the echoes retrieved from the cylinder with the screw, it can be noted in Fig. 4.4 that the first echo is greater in amplitude for the screw than the cylinder. This might be a cause from destructive interference with the path seen in Fig. 2.4c, which would not have as big impact for the screws where the threads suppress sidewall

References

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